In this work, we present a comprehensive framework for fusing
measurements from multiple and generally placed accelerometers and
gyroscopes to perform
inertial navigation. The traditional
inertial navigation equations are extended to include the angular acceleration
provided by the
accelerometer array. The angular acceleration admits more accurate numerical
integration of the orientation with second-order accuracy compared to
the traditional
first-order accuracy that can be achieved when only using the
gyroscopes [1]}.
Since orientation errors are the
most significant error source in inertial navigation,
improving the orientation estimation will reduce the overall
navigation error. To practically benefit from the angular
acceleration, estimated by the accelerometer array, requires prior
knowledge of the inertial sensor
array. Therefore, we propose four state-space models, a.k.a
mechanization equations, using different underlying assumptions
regarding the time propagation of the orientation state. We evaluate
these models using a
Lie group Kalman filter through simulations and
experimentsReproducible research: The data and the code used
in the simulations and the experiments are available at
https://github.com/hcarlsso/Array-IN.
Using the
Kalman filter, we show that the non linear problem
of fusing the measurements
of multiple accelerometers and gyroscopes can be
solved by Kalman updates instead of using an iterative optimization
method [2]}. Moreover, the Kalman updates are also shown
to stabilize the potentially unstable ODE
when integrating the angular velocity. Finally, we note that
individual accelerometer biases
in an accelerometer array are unobservable [3]}. However,
we show that the individual accelerometer biases and the joint accelerometer
covariance matrix can be
replaced by a six-dimensional bias term and a six-by-six covariance
matrix. The dimensions of these new terms are fixed and
independent of the number of accelerometers.
